Circumbinary planets in coplanar triple-star systems: I. Minimum eccentricity variation region
We aim to determine the conditions under which Lidov-Kozai oscillations can arise for circumbinary planets in hierarchical triple-star systems, and to identify the orbital regions where competing secular perturbations minimize eccentricity excitation. We analyzed the statistical properties of hierarchical triples from the Multiple Star Catalog to construct representative synthetic configurations. We then performed N-body simulations of circumbinary planets embedded in these systems, systematically exploring their orbital evolution and long-term stability across a range of semimajor axes and inclinations. For representative compact hierarchical triple-star systems with masses $m_0 \sim m_1 \approx 0.75\,{\rm M}_\odot$ and $m_2 \approx 0.6\,{\rm M}_\odot$ and eccentricities $e_1 \sim 0.1$ and $e_2 \sim 0.4$, dynamically significant stable circumbinary regions exist only in sufficiently hierarchical configurations, with period ratios $P_2 /P_1 \gtrsim 10^2$. In this regime, the secular competition between the tertiary perturbations and the apsidal precession induced by the compact inner binary determines both the onset of Lidov-Kozai oscillations beyond a critical distance ($a_{\rm LK}$) and the location of a region where eccentricity variations are minimized, i.e., the minimum eccentricity variation region (MER). We derived analytical estimates for these dynamical features, showing that the MER is well described by the analytical quantity $a_{\rm short}$ on short and intermediate timescales, while on secular timescales it converges toward the equilibrium prediction $a_{\rm ME}$. These analytical estimates agree well with N-body simulations. Application to the observed triple system WDS 08403+1921 confirms that $a_{\rm LK}$ and $a_{\rm ME}$ accurately identify the main dynamical features of the stability map.